Quantum treatment of phonon scattering for modeling of three-dimensional atomistic transport

Based on the nonequilibrium Green's function formalism, we show a numerically efficient method to treat inelastic scattering in multidimensional atomistic codes. Using a simple rescaling approach, we detail the calculations of the lowest-order approximation (LOA) [Y. Lee et al., Phys. Rev. B 93, 205411 (2016)] series to the usual, computationally intensive, self-consistent Born approximation (SCBA). This, combined with the analytic continuation technique of Padé approximants, is applied to an atomistic code based on a tight-binding $s{p}^3{d}^5{s}^{*}$ model for electrons and holes, and a modified valence-force-field method for phonons. Currents in Si and Ge gate-all-around nanowire transistors are then computed considering the main crystallographic transport directions ($\langle 100\rangle$, $\langle 110\rangle$, $\langle 111\rangle$) for both $n$-type and $p$-type devices. Our results show that in most configurations, third-order LOA currents are enough to achieve a high agreement with SCBA results, while reducing the calculation time by about one order. In addition, we propose a criterion to determine the validity of such expansion techniques.

Figure 1

Schematic representation of the silicon GAA NW transistors considered in this work (with a $\langle 100\rangle$ crystallographic orientation in the present case). The length of the gate is $L_G=5\mathrm{nm}$, the source and drain extensions $L_{S/D}=15\mathrm{nm}$, and the diameter of NW $d_{\mathrm{NW}}=3\mathrm{nm}$. Source and drain regions are doped with a doping concentration of donors or acceptors $N_{D/A}=1\times {10}^{20}{\mathrm{cm}}^{-3}$. $\mathrm{Si}\langle 110\rangle$, $\mathrm{Si}\langle 111\rangle$, and $\mathrm{Ge}\langle 110\rangle$ are also considered (not shown).

Figure 2

Rescaled Green's functions, (a) $G_1^{{\lambda }_1}$ of first SCBA iteration and (b) $G_2^{{\lambda }_2}$ of second SCBA iteration, described by Feynman diagrams. Conserving and nonconserving terms are arranged in ascending order in interactions. Thin lines with an arrow indicate electron noninteracting Green's function. Dashed lines are for free phonon propagators.